# Full text of "Treatise On Analysis Vol-Ii"

## See other formats

16 BAIRE'S THEOREM AND ITS CONSEQUENCES 95 20. Let Z be a complete metric space and / a weakly continuous mapping of Z into a separable Hilbert space E. (a) If A is any closed convex set in E, show that /"HA) is closed in Z (Cf. Section 12.15, Problem 3(b)). (b) For each integer n > 1, let Dn be the set of points z e Z such that the oscillation of the function / (considered as a mapping of a metric space into a metric space) is ;>!/«. The set Dn is closed (Section 12.7, Problem 4). Show that Dw is meager in Z. (Cover E by a sequence (Sk) of closed balls of diameter < 1/2/z, and use (a) and Baire's theorem to show that Dn n/-1(S*) is a nowhere dense closed set in Z, for each k.) (c) Deduce from (a) that there exists a meager set M in Z such that / is strongly continuous at every point of Z — M. 21. Let E = JT(R) be the subspace consisting of continuous functions with compact support, in the Banach space ^(R) of real-valued bounded functions on R. Let u» n be the continuous linear form /(-»£/(/:) on E. Show that the sequence (un) converges *=i simply on E but is not bounded in the dual E', and that its limit u is not continuous on E. 22. Let E be the subspace consisting of sequences (£„) with finite support, in the Hilbert space /2, so that E is the set of all (finite) linear combinations of the vectors en in I2 (6.5). Let u be the endomorphism of E for which u(en) = nen for all n. Show that, for every y e E, the linear form x\- *(u(x)\y) is continuous on E, but that u is not a con- tinuous endomorphism of E. 23. Let E be the subset of R2 consisting of the line D = {0} x R and the points (I In, kin2), where n and k belong to Z and n > 0. (a) For every point (0, y) e E and every integer n > 0, let Tn(y) be the set of points (u, u) e E such that u^l/n and \v — y\<Zu. Show that if we take the sets Tn(y) (n^l) as a fundamental system of neighborhoods of (0, y), for each y e R, and if for each other point z of E we take the set consisting of that point alone as a fundamental system of neighborhoods of z, then we have a HausdorfT topology 9~ on E, for which each subspace Tn(y) is metrizable and compact. The topology induced by F on D is the discrete topology. (b) Let A be the set {0} x Q, which is closed in D and whose complement B in D is also closed, If a set U which is open with respect to & contains B, show that there exists an interval {0} x [a, b] in D (with a < b) and an integer n such that U contains the union of the sets Tn(y) for y e [a, b] and (0, y) e B. (Use the fact that Q is a meager subset of R in the usual topology.) Deduce that every neighborhood of A in E meets U, and hence that the space E is not metrizable, although it contains a discrete denumerable dense subset (use (4.5.2)). 24. Let (an) be a sequence of real numbers. Suppose that, for each sequence (£„) of real numbers belonging to the space /* (Section 5.7, Problem 1) (resp. to the space /2 (6.5)), the series with general term an£n is convergent. Prove that the sequence (#„) belongs to /°° (Section 5.6, Problem 1) (resp, belongs to /2). (For each integer N, N consider the continuous linear form x i— > JT an $ n on /* (resp. /2), and apply the Banach- ir»l Steinhaus theorem.) and a